Stochastic Averaging of The Einstein Vacuum Equations on a Toroidal Manifold with Randomly Perturbed Radial Moduli: Stability Criteria and Induced 'Cosmological Constant' Terms

Abstract

The Einstein vacuum equations on an (n+1)-dimensional toroidal manifold $\mathbb{M}^{n+1}=\mathbb{T}^{n}\times\mathbb{R}^{+}$ reduce to a system of n-dimensional nonlinear ODEs in terms of the set of toroidal radii $(a_{i})_{i=1}^{n}$ or the radial moduli fields $(\psi_{i})_{i=1}^{n}=(\log(a_{i}(t))_{i=1}^{n}$ of the n-torus $\mathbb{T}^{n}$. This geometry is also the basis of Kasner-Bianchi-type cosmologies. The equations are trivially satisfied for static solutions $\psi_{i}^{E}=\psi^{E}$ or radii $a_{i}^{E}=a^{E}$, describing an initially static toroidal 'micro-universe' or 'vacuum bubble'. It is Lyapunov stable to short-pulse deterministic perturbations, which have a sharp Gaussian profile: the perturbed radii rapidly converge to new attractors and therefore to new stable equilibria. These perturbations induce transitions between stable states. Introducing classical random fluctuations or perturbations, with a regulated covariance, the radial moduli become Gaussian random fields paramatrizing a 'toroidal random geometry'. The randomly perturbed Einstein equations are then interpreted as a stochastic n-dimensional nonlinear dynamical system. Non-vanishing 'cosmological constant' terms are retained within the averaged equations since they are nonlinear. This is analogous to averaging the Navier-Stokes equations in statistical turbulence theory, which yields an additional non-vanishing Reynolds term, since like the Einstein equations they are also of nonlinear hyperbolic type. The expectations of the randomly perturbed toric radii can be estimated from a cumulant expansion method. The initially static toroidal vacuum bubble undergoes eternal 'noise-induced' stochastic exponential growth or inflation. Random radial moduli fields within this scenario therefore act like a 'dark energy'. Finally, a class of random perturbations is considered for which this Einstein system is stable.